Determining the angle of arrival of a target signal received by an array of antenna elements

ABSTRACT

In a system for determining the angle of arrival of a target signal received by an array of antenna elements, a pair of receivers simultaneously obtain observations of a received target signal from multiple elements of an array of antenna elements; and a computer processes the simultaneously obtained samples of the target signal to determine a maximum likelihood estimation (MLE) of the angle of arrival φ of the target signal by using the following equation: φ MLE =argmax φ Re(α*β). The value of β is determined in accordance with whether the target signal is known or unknown. When the target signal is unknown, the computer also processes the simultaneously obtained samples of the target signal to estimate the bandwidth of the received target signal by using binary hypotheses and a generalized log likelihood ratio test (GLLRT) or by using multiple hypotheses and pair-wise generalized log likelihood ratio tests. The value of a bandwidth constraint M that is associated with the estimated bandwidth is used to derive the value of β that is used to determine the MLE of φ.

BACKGROUND OF THE INVENTION

The present invention generally pertains to determining the angle ofarrival (AOA) of a target signal received by an array of antennaelements.

Arrays of antenna elements are commonly used in a system for estimatingthe AOA of a received target signal. For tactical signal-interceptapplications it is desirable for the signal-intercept hardware to be ofminimal size, weight, and power (SWAP). To realize minimal SWAP it isdesirable to use a single receiver and to commutate the antenna elementsof the array.

SUMMARY OF THE INVENTION

The present invention provides a method of estimating the angle ofarrival of a target signal received by an array of antenna elements,comprising the steps of:

(a) with a pair of receivers, simultaneously obtaining samples of areceived target signal from multiple elements of an array of antennaelements; and

(b) with a computer, processing the simultaneously obtained samples ofthe target signal to determine a maximum likelihood estimation (MLE) ofthe angle of arrival φ of the target signal by using the followingequation:φ_(MLE)=argmax_(φ) Re(α*β)wherein α is a complex vector that represents a phase differenceassociated with the angle of arrival that should be observed uponreceipt of the signal by two particular antenna elements from which thesamples are obtained; andβ represents the phase difference that is observed upon receipt of thesignal by the two particular antenna elements from which the samples areobtained;

wherein when the target signal is unknown, β_(n)=y*_(2n−1)y_(2n) in thetime domain and β_(n)=Y*_(2n−1)Y_(2n) in the frequency domain,

wherein y_(2n−1) and y_(2n) are complex N-tuple vectors representing thesamples obtained from the nth simultaneously sampled pair of antennaelements and Y is a Fourier transform of y.

The present invention also provides a method of estimating the angle ofarrival of a target signal received by an array of antenna elements,comprising the steps of:

(a) with a pair of receivers, simultaneously obtaining samples of areceived target signal from multiple elements of an array of antennaelements; and

(b) with a computer, processing the simultaneously obtained samples ofthe target signal to determine a maximum likelihood estimation (MLE) ofthe angle of arrival φ of the target signal by using the followingequation:φ_(MLE)=argmax_(φ) Re(α*β)wherein α is a complex vector that represents a phase differenceassociated with the angle of arrival that should be observed uponreceipt of the signal by two particular antenna elements from which thesamples are obtained; andβ represents the phase difference that is observed upon receipt of thesignal by the two particular antenna elements from which the samples areobtained;

wherein when the target signal is known,

$\beta_{n} = \frac{\left( {x_{n}^{*}y_{{2n} - 1}} \right)*\left( {x_{n}^{*}y_{2n}} \right)}{{x_{n}}^{2}}$in the time domain and

$\beta_{n} = \frac{\left( {X_{n}^{*}Y_{{2n} - 1}} \right)*\left( {X_{n}^{*}Y_{2n}} \right)}{{X_{n}}^{2}}$in the frequency domain,wherein X and Y are Fourier transforms of x and y respectively,wherein x_(n) and X_(n) are complex N-tuple vectors representing theknown target signal in the time domain and in the frequency domainrespectively; and

wherein y_(2n−1) and y_(2n) are complex N-tuple vectors representing thesamples obtained from the nth simultaneously sampled pair of antennaelements and Y is a Fourier transform of y.

The present invention further provides a method of estimating thebandwidth of a target signal received by an array of antenna elements,comprising the steps of:

(a) with a pair of receivers, simultaneously obtaining samples of areceived target signal from multiple elements of an array of antennaelements; and

(b) with a computer, processing the simultaneously obtained samples ofthe target signal to estimate the bandwidth of the received targetsignal by using binary hypotheses and a generalized log likelihood ratiotest (GLLRT).

The present invention still further provides a method of estimating thebandwidth of a target signal received by an array of antenna elements,comprising the steps of:

(a) with a pair of receivers, simultaneously obtaining samples of areceived target signal from multiple elements of an array of antennaelements; and

(b) with a computer, processing the simultaneously obtained samples ofthe target signal to estimate the bandwidth of the received targetsignal by using multiple hypotheses and pair wise generalized loglikelihood ratio tests in accordance with:

$\left. \begin{matrix}{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{2} \right)}}^{2}} < {\lambda\left( {M_{1},M_{2}} \right)}} \\{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{3} \right)}}^{2}} < {\lambda\left( {M_{1},M_{3}} \right)}}\end{matrix}\Rightarrow M_{1} \right.$wherein M specifies a bandwidth constraint expressed by a set of tuplesof X(φ) whereX(φ) may be non-zero,X(φ) is an estimate of the Fourier transform of the unknown targetsignal,Y is a Fourier transform of the sample obtained from the sampled antennaelement, and

E(M)² =  < Y₁, Y₁ > + < Y₂, Y₂ > −c( < Y₁, Y₁>_(M)+2 < Y₁, Y₂>_(M)+ < Y₂, Y₂>_(M)).

The present invention additionally provides systems for performing theabove-described methods and computer readable storage media includingcomputer executable program instructions for causing one or morecomputers to perform and/or enable the steps of the respectiveabove-described methods.

Additional features of the present invention are described withreference to the detailed description.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a block diagram of an exemplary system in which the methods ofthe present invention are performed.

FIG. 2 is a diagram generally showing an embodiment of a method of thepresent invention that is used when the target signal is unknown.

FIG. 3 is a diagram generally showing an embodiment of a method of thepresent invention that is used when the target signal is known.

DETAILED DESCRIPTION

Referring to FIG. 1, an exemplary system in which the methods of thepresent invention are performed includes an array of antenna elements10, a pair of receivers 12 and a computer 14. The computer 14 contains adigital signal processor and computer readable storage media thatincludes computer executable program instructions for causing thecomputer to perform and/or enable the various processing steps that aredescribed herein. These instructions are stored in the computer readablestorage media of the computer when the computer is manufactured and/orupon being downloaded via the Internet or from a portable computerreadable storage media containing such instructions.

Samples 16 of a received target signal are simultaneously obtained bythe pair of receivers 12 from multiple elements of an array of antennaelements 10 (as shown at 20 in FIG. 2 and at 26 in FIG. 3); and thecomputer 14 is adapted for processing the simultaneously obtainedsamples 16 of the target signal (as shown at 22 in FIG. 2 and at 28 inFIG. 3) to determine a maximum likelihood estimation (MLE) of the angleof arrival φ of the target signal by using the following equation:φ_(MLE)=argmax_(φ) Re(α*β)   [Eq. 1]α is a complex vector that represents a phase difference associated withthe angle of arrival that should be observed upon receipt of the signalby two particular antenna elements from which the samples are obtained;and β represents the phase difference that is observed upon receipt ofthe signal by the two particular antenna elements from which the samplesare obtained. The symbol * is a Hermitian operator.

When the target signal is unknown, in the time domain,β_(n) =y* _(2n−1) y _(2n)   [Eq. 2]and in the frequency domain,β_(n) =Y* _(2n−1) Y _(2n)   [Eq. 3]y_(2n−1) and y_(2n) are complex N-tuple vectors representing the samplesobtained from the nth simultaneously sampled pair of antenna elementsand Y is a Fourier transform of y. In a preferred embodiment,

When the target signal is known, in the time domain,

$\begin{matrix}{\beta_{n} = \frac{\left( {x_{n}^{*}y_{{2n} - 1}} \right)*\left( {x_{n}^{*}y_{2n}} \right)}{{x_{n}}^{2}}} & \left\lbrack {{Eq}.\mspace{14mu} 4} \right\rbrack\end{matrix}$and in the frequency domain,

$\begin{matrix}{\beta_{n} = \frac{\left( {X_{n}^{*}Y_{{2n} - 1}} \right)*\left( {X_{n}^{*}Y_{2n}} \right)}{{X_{n}}^{2}}} & \left\lbrack {{Eq}.\mspace{14mu} 5} \right\rbrack\end{matrix}$X and Y are Fourier transforms of x and y respectively, and x_(n) andX_(n) are complex N-tuple vectors representing the known target signalin the time domain and in the frequency domain respectively. y_(2n−1)and y_(2n) are complex N-tuple vectors representing the samples obtainedfrom the nth simultaneously sampled pair of antenna elements.

For a phased array of antenna elements, whether or not the target signalis known,α_(n) =e ^(i(μ) ^(2n) ^((φ)−μ) ^(2n−1) ^((φ)))  [Eq. 6]μ is a function of the angle of arrival that depends upon the geometryof the array of antenna elements.

Referring to FIG. 2, the computer 14 is also adapted for processing thesimultaneously obtained samples 16 of the target signal when the targetsignal is unknown to estimate the bandwidth of the target signal, asshown at 24.

In one embodiment, the computer 14 is adapted for estimating thebandwidth of the received target signal by using binary hypotheses and ageneralized log likelihood ratio test (GLLRT):

$\begin{matrix}{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{2} \right)}}^{2}}\overset{M_{1}}{\underset{M_{2}}{\lessgtr}}{\lambda\left( {M_{1},M_{2}} \right)}} & \left\lbrack {{Eq}.\mspace{14mu} 7} \right\rbrack\end{matrix}$λ(M₁, M₂) is an appropriately chosen constant threshold. M specifies abandwidth constraint expressed by a set of tuples of X(φ) where X(φ) maybe non-zero. X(φ) is an estimate of the Fourier transform of the unknowntarget signal. Y is a Fourier transform of the sample obtained from thesampled antenna element.

$\begin{matrix}{{{{E(M)}}^{2} = {< Y_{1}}},{Y_{1} > {+ {< Y_{2}}}},{Y_{2} > {- {c\left( {{< Y_{1}},{Y_{1} >_{M}{{{+ 2}{{{< Y_{1}},{Y_{2} >_{M}}}}} +} < Y_{2}},{Y_{2} >_{M}}} \right)}}}} & \left\lbrack {{Eq}.\mspace{14mu} 8} \right\rbrack\end{matrix}$

In another embodiment, the computer 14 is adapted for estimating thebandwidth of the received target signal by using multiple hypotheses andpair wise generalized log likelihood ratio tests in accordance with:

$\begin{matrix}\left. \begin{matrix}{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{2} \right)}}^{2}} < {\lambda\left( {M_{1},M_{2}} \right)}} \\{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{3} \right)}}^{2}} < {\lambda\left( {M_{1},M_{3}} \right)}}\end{matrix}\Rightarrow M_{1} \right. & \left\lbrack {{Eq}.\mspace{14mu} 9} \right\rbrack\end{matrix}$M specifies a bandwidth constraint expressed by a set of tuples of X(φ)where X(φ) may be non-zero. X(φ) is an estimate of the Fourier transformof the unknown target signal. Y is a Fourier transform of the sampleobtained from the sampled antenna element.

$\begin{matrix}{{{{E(M)}}^{2} = {< Y_{1}}},{Y_{1} > {+ {< Y_{2}}}},{{Y_{2} > {- {c\left( {{< Y_{1}},{Y_{1} >_{M}{{{+ 2}{{{< Y_{1}},{Y_{2} >_{M}}}}} +} < Y_{2}},{Y_{2} >_{M}}} \right)}}};}} & \left\lbrack {{Eq}.\mspace{14mu} 8} \right\rbrack\end{matrix}$

The computer 14 is further adapted for deriving the value ofβ_(n)=Y*_(2n−1)Y_(2n) in the frequency domain or the value ofβ_(n)=<Y_(2n−1), Y_(2n)>_(M) in the frequency domain or the value ofβ_(n)=<y_(2n−1), y_(2n)>_(M) in the time domain by using the respectivevalue M that is associated with the estimated bandwidth pursuant to theapplicable GLLRT or GLLRTs.

Referring to FIG. 3, when the target signal is known, samples of thetarget signal are obtained as shown at 28, and the computer 14 processesthe samples 16 in accordance with above-recited Equations 1, 4 and 5 todetermine the MLE of the angle of arrival φ of the target signal.

A discussion of the applicability of various equations to differentembodiments of the present invention follows.

Single Pair of Antenna Elements and an Unknown Target Signal

Consider first the estimation of an MLE for the AOA with two receiverswhen the target signal is unknown. Suppose two antenna elements aresampled simultaneously, and N samples are observed on each element. Letthe complex N-tuple vectors y₁ and y₂ denote the samples observed on thefirst element and second element, respectively. Then,

$\begin{matrix}{\begin{pmatrix}y_{1} \\y_{2}\end{pmatrix} = {{\begin{pmatrix}{{\mathbb{e}}^{{\mathbb{i}\mu}_{1}{(\phi)}}I} \\{{\mathbb{e}}^{{\mathbb{i}\mu}_{2}{(\phi)}}I}\end{pmatrix}x} + \begin{pmatrix}v_{1} \\v_{2}\end{pmatrix}}} & \left\lbrack {{Eq}.\mspace{14mu} 10} \right\rbrack\end{matrix}$The vector x denotes the unknown target signal. e^(iμ) ¹ ^((φ)) ande^(iμ) ² ^((φ)) are unit modulus complex gains on each antenna elementthat are related to the AOA φ. For a uniform circular array of antennaelements, μ_(k)(φ)=2πr/λsin(φ−θ_(k)). Any non-unit modulus of the gainmay be regarded as absorbed into x. I is the compatible identity matrix.The vector v_(k) is complex additive white Gaussian noise (AWGN), andthe noise from each antenna element is assumed to be independent.For AWGN the MLE for φ is given by,

$\begin{matrix}{\phi_{MLE} = {{argmin}_{\phi}{{\begin{pmatrix}y_{1} \\y_{2}\end{pmatrix} - {\begin{pmatrix}{{\mathbb{e}}^{{\mathbb{i}\mu}_{1}{(\phi)}}I} \\{{\mathbb{e}}^{{\mathbb{i}\mu}_{2}{(\phi)}}I}\end{pmatrix}{x(\phi)}}}}^{2}}} & \left\lbrack {{Eq}.\mspace{14mu} 11} \right\rbrack\end{matrix}$where x(φ) is the least-squares optimal estimate of x and is dependenton φ.Equation 11 is a time-domain problem. Alternatively, this may bereformulated in the frequency domain, as follows. Let W_(N) denote thelinear transformation associated with an N-point DFT, and let

$\begin{matrix}{{W\; 2_{N}} = \begin{pmatrix}W_{N} & 0 \\0 & W_{N}\end{pmatrix}} & \left\lbrack {{Eq}.\mspace{14mu} 12} \right\rbrack\end{matrix}$Note that W2_(N)≠W_(2N). First, W2_(N) is unitary (W2_(N)*W2_(N)=I), asW_(N) is unitary. Hence, multiplication by W2_(N) does not change thelength of a vector, and Equation 12 may be rewritten as,

$\begin{matrix}{\phi_{MLE} = {{argmin}_{\phi}{{W\; 2_{N}\left\{ {\begin{pmatrix}y_{1} \\y_{2}\end{pmatrix} - {\begin{pmatrix}{{\mathbb{e}}^{{\mathbb{i}\mu}_{1}{(\phi)}}I} \\{{\mathbb{e}}^{{\mathbb{i}\mu}_{2}{(\phi)}}I}\end{pmatrix}{x(\phi)}}} \right\}}}^{2}}} & \left\lbrack {{Eq}.\mspace{14mu} 13} \right\rbrack\end{matrix}$Secondly,

$\begin{matrix}{{{W\; 2_{N}\begin{pmatrix}y_{1} \\y_{2}\end{pmatrix}} = {\begin{pmatrix}{W_{N}y_{1}} \\{W_{N}y_{2}}\end{pmatrix} = \begin{pmatrix}Y_{1} \\Y_{2}\end{pmatrix}}},} & \left\lbrack {{Eq}.\mspace{14mu} 14} \right\rbrack\end{matrix}$where Y_(k) is the N-point DFT of y_(k) (i.e. U denotes the Fouriertransform of u). Therefore, Equation 13 may be rewritten as,

$\begin{matrix}{\phi_{MLE} = {{argmin}_{\phi}{{\begin{pmatrix}Y_{1} \\Y_{2}\end{pmatrix} - {\begin{pmatrix}{{\mathbb{e}}^{{\mathbb{i}\mu}_{1}{(\phi)}}I} \\{{\mathbb{e}}^{{\mathbb{i}\mu}_{2}{(\phi)}}I}\end{pmatrix}{X(\phi)}}}}^{2}}} & \left\lbrack {{Eq}.\mspace{14mu} 15} \right\rbrack\end{matrix}$Note that

$\begin{matrix}{{\begin{pmatrix}{{\mathbb{e}}^{{\mathbb{i}\mu}_{1}{(\phi)}}I} \\{{\mathbb{e}}^{{\mathbb{i}\mu}_{2}{(\phi)}}I}\end{pmatrix}{x(\phi)}} = \begin{pmatrix}{{\mathbb{e}}^{{\mathbb{i}\mu}_{1}{(\phi)}}{x(\phi)}} \\{{\mathbb{e}}^{{\mathbb{i}\mu}_{2}{(\phi)}}{x(\phi)}}\end{pmatrix}} & \left\lbrack {{Eq}.\mspace{14mu} 16} \right\rbrack\end{matrix}$Equation 15 is a least-squares problem. Let,

$\begin{matrix}{{A(\phi)} = \begin{pmatrix}{{\mathbb{e}}^{{\mathbb{i}\mu}_{1}{(\phi)}}I} \\{{\mathbb{e}}^{{\mathbb{i}\mu}_{2}{(\phi)}}I}\end{pmatrix}} & \left\lbrack {{Eq}.\mspace{14mu} 17} \right\rbrack\end{matrix}$Then Equation 17 may be rewritten as,φ_(MLE)=argmin_(φ) ∥Y−A(φ)X(φ)∥²   [Eq. 18]It is a standard result from linear algebra that X(φ) is given by,A(φ)*A(φ)X(φ)=A(φ)*Y   [Eq. 19]Since A(φ) is orthogonal,X(φ)=cA(φ)*Y   [Eq. 20]for some real constant c.As the optimal error vector is orthogonal to the column space of A(φ),∥Y−A(φ)X(φ)∥² =∥Y∥ ² −Y*A(φ)X(φ)   [Eq. 21]and Equation 18 may be rewritten as,φ_(MLE)=argmax_(φ) Y*A(φ)X(φ)   [Eq. 22]orφ_(MLE)=argmax_(φ) ∥A(φ)*Y∥ ²   [Eq. 23]In Equation 23,

$\begin{matrix}{{{{A(\phi)}^{*}Y}}^{2} = {\sum\limits_{k}^{\;}\;{{{{\mathbb{e}}^{- {{\mathbb{i}\mu}_{1}{(\phi)}}}Y_{1,k}} + {{\mathbb{e}}^{- {{\mathbb{i}\mu}_{2}{(\phi)}}}Y_{2,k}}}}^{2}}} & \left\lbrack {{Eq}.\mspace{14mu} 24} \right\rbrack\end{matrix}$and|e ^(−iμ) ¹ ^((φ)) Y _(1,k) +e ^(−iμ) ² ^((φ)) Y _(2,k)|² =|Y_(1,k)|²+2Re{(e ^(−iμ) ¹ ^((φ)) Y _(1,k))*(e ^(−iμ) ² ^((φ)) Y_(2,k))}+|Y _(2,k)|²   [Eq. 25]Therefore, Equation 23 may be rewritten as,

$\begin{matrix}\begin{matrix}{\phi_{MLE} = {{argmax}_{\phi}{Re}\left\{ {{\mathbb{e}}^{{\mathbb{i}}{({{\mu_{1}{(\phi)}} - {\mu_{2}{(\phi)}}})}}{\sum\limits_{k}^{\;}\;{Y_{1,k}^{*}Y_{2,k}}}} \right\}}} \\{= {{argmax}_{\phi}{Re}\left\{ {{\mathbb{e}}^{{\mathbb{i}}{({{\mu_{1}{(\phi)}} - {\mu_{2}{(\phi)}}})}}Y_{1}^{*}Y_{2}} \right\}}}\end{matrix} & \left\lbrack {{Eq}.\mspace{14mu} 26} \right\rbrack\end{matrix}$

It follows that Y*₁Y₂ is a sufficient statistic for computing φ_(MLE).(The vectors Y₁ and Y₂ may be reduced to the complex scalar Y*₁Y₂without any loss of performance.

Note that

$\begin{matrix}{{Y_{1}^{*}Y_{2}} \neq {\sum\limits_{k}^{\;}\;{\frac{Y_{2,k}}{Y_{1,k}}.}}} & \left\lbrack {{Eq}.\mspace{14mu} 27} \right\rbrack\end{matrix}$

There is a clear duality between the time and frequency domainformulations. Equation 15 is a frequency-domain formulation of thisproblem, and Equation 26 yields the associated solution for φ_(MLE). Thetime-domain formulation in Equation 11 is structurally identical toEquation 15, so it follows immediately that there is also a time-domainsolution given by,φ_(MLE)=argmax_(φ) Re{e ^(i(μ) ¹ ^((φ)−μ) ² ^((φ))) y* ₁ y ₂}  [Eq. 28]

The frequency-domain approach has the immediate advantage that manysignals are localized in frequency and noise may be easily eliminated.Of course, one may use a Fourier transform and inverse Fourier transformto eliminate noise in the frequency domain, but solve the problem in thetime domain using Equation 28. The latter approach has the disadvantageof require two transforms, so the frequency-domain approach ispreferable.

In comparison with algorithms for the single receiver problem, the tworeceiver algorithm is considerably simpler. In the unknown target signalscenario, this simplicity results from the fact that the inner-productin the sufficient statistic cancels the angular component of the unknownsignal.

Single Pair of Antenna Elements and Known Target Signal

Consider the construction of an MLE for the AOA with two receivers wherethe target signal is known.

$\begin{matrix}{\begin{pmatrix}y_{1} \\y_{2}\end{pmatrix} = {{\begin{pmatrix}{{\mathbb{e}}^{{\mathbb{i}}_{\mu_{1}{(\phi)}}}x} \\{{\mathbb{e}}^{{\mathbb{i}}_{\mu_{2}{(\phi)}}}x}\end{pmatrix}z} + \begin{pmatrix}v_{1} \\v_{2}\end{pmatrix}}} & \left\lbrack {{Eq}.\mspace{14mu} 29} \right\rbrack\end{matrix}$This equation is entirely similar to Equation 10. The only difference isthe introduction of complex scalar z. This permits uncertainty of theamplitude and phase of x which is otherwise known, and is a morepractical model. Note that

$\quad\begin{pmatrix}{{\mathbb{e}}^{{\mathbb{i}}_{\mu_{1}{(\phi)}}}x} \\{{\mathbb{e}}^{{\mathbb{i}}_{\mu_{2}{(\phi)}}}x}\end{pmatrix}$is a vector.In the same manner as the previous development of Equation 15,

$\begin{matrix}{\phi_{MLE} = {{argmin}_{\phi}{{\begin{pmatrix}Y_{1} \\Y_{2}\end{pmatrix} - {\begin{pmatrix}{{\mathbb{e}}^{{\mathbb{i}}_{\mu_{1}{(\phi)}}}X} \\{{\mathbb{e}}^{{\mathbb{i}}_{\mu_{2}{(\phi)}}}X}\end{pmatrix}z}}}^{2}}} & \left\lbrack {{Eq}.\mspace{14mu} 30} \right\rbrack\end{matrix}$and in the same manner as the previous development of Equation 23,φ_(MLE)=argmax_(φ) ∥a(φ)*Y∥ ²   [Eq. 31]with

$\begin{matrix}{{a(\phi)} = \begin{pmatrix}{{\mathbb{e}}^{{\mathbb{i}}_{\mu_{1}{(\phi)}}}X} \\{{\mathbb{e}}^{{\mathbb{i}}_{\mu_{2}{(\phi)}}}X}\end{pmatrix}} & \left\lbrack {{Eq}.\mspace{14mu} 32} \right\rbrack\end{matrix}$In Equation 31,∥a(φ)*Y∥ ² =|e ^(−iμ) ¹ ^((φ)) X*Y ₁ +e ^(−iμ) ² ^((φ)) X*Y ₂|²   [Eq.33]and,|e ^(−iμ) ¹ ^((φ)) X*Y ₁ +e ^(−iμ) ² ^((φ)) X*Y ₂|² =|X*Y₁|²+2Re{(e ^(−iμ) ¹ ^((φ)) X*Y ₁)*e ^(−iμ) ² ^((φ)) X*Y ₂ }+|X*Y ₂|²  [Eq. 34]Therefore, Equation 31 may be rewritten as,φ_(MLE)=argmax_(φ) Re{e ^(i(μ) ¹ ^((φ)−μ) ² ^((φ)))(X*Y ₁)*X*Y ₂}  [Eq.35]X*Y₁ and X*Y₂ are sufficient statistics for computing φ_(MLE).As before, a time-domain formulation is analogous,φ_(MLE)=argmax_(φ) Re{e ^(i(μ) ¹ ^((φ)−μ) ² ^((φ)))(x*y ₁)*x*y ₂}  [Eq.36]x*y₁ and x*y₂ are sufficient statistics for computing φ_(MLE).

In general, in the argument of the argmax function in Equation 31 theexpression ∥a(φ)∥² would appear as a divisor. Here, this expression doesnot depend on φ and is ignored. This expression does however depend onX. With only a single element pair, this again may be ignored. Withmultiple element pairs as discussed subsequently, this must beconsidered. This is the reason for the subsequent divisions by |x|² or|X|².

Multiple Pairs of Antenna Elements

For an unknown target signal, the extension to multiple pairs ofelements may be constructed as follows. Refer to Equation 10. With twopairs of elements this may be written as,

$\begin{matrix}{\begin{pmatrix}y_{12} \\y_{34}\end{pmatrix} = {{\begin{pmatrix}{{\mathbb{e}}^{{\mathbb{i}}_{\mu_{1}{(\phi)}}}I} & 0 \\{{\mathbb{e}}^{{\mathbb{i}}_{\mu_{2}{(\phi)}}}I} & \; \\\; & {{\mathbb{e}}^{{\mathbb{i}}_{\mu_{3}{(\phi)}}}I} \\0 & {{\mathbb{e}}^{{\mathbb{i}}_{\mu_{4}{(\phi)}}}I}\end{pmatrix}\begin{pmatrix}x_{12} \\x_{34}\end{pmatrix}} + \begin{pmatrix}v_{12} \\v_{34}\end{pmatrix}}} & \left\lbrack {{Eq}.\mspace{14mu} 37} \right\rbrack\end{matrix}$where

$y_{12} = {\begin{pmatrix}y_{1} \\y_{2}\end{pmatrix}.}$Equation 37 may be written more compactly as,y=A(φ)x+v   [Eq. 38]Then,φ_(MLE)=argmax_(φ) ∥A(φ)*y∥ ²   [Eq. 39]as A(φ) is orthogonal.φ_(MLE)=argmax_(φ) {∥A ₁₂(φ)*y ₁₂∥² +∥A ₃₄(φ)*y ₃₄∥²}  [Eq. 40]where

${A_{12}(\phi)} = \begin{pmatrix}{{\mathbb{e}}^{{\mathbb{i}}_{\mu_{1}{(\phi)}}}I} \\{{\mathbb{e}}^{{\mathbb{i}}_{\mu_{2}{(\phi)}}}I}\end{pmatrix}$and

${A_{34}(\phi)} = {\begin{pmatrix}{{\mathbb{e}}^{{\mathbb{i}}_{\mu_{3}{(\phi)}}}I} \\{{\mathbb{e}}^{{\mathbb{i}}_{\mu_{4}{(\phi)}}}I}\end{pmatrix}.}$From previous arguments,φ_(MLE)=argmax_(φ) Re{e ^(i(μ) ¹ ^((φ)−μ) ² ^((φ))) y* ₁ y ₂ +e ^(i(μ) ³^((φ)−μ) ⁴ ^((φ))) y* ₃ y ₄}  [Eq. 41]With an arbitrary number of pairs of elements,

$\begin{matrix}{\phi_{MLE} = {{argmin}_{\phi}{\sum\limits_{n}{{Re}\left\{ {{\mathbb{e}}^{{\mathbb{i}}_{({{\mu_{{2n} - 1}{(\phi)}} - {\mu_{2n}{(\phi)}}})}}y_{{2n} - 1}^{*}y_{2n}} \right\}}}}} & \left\lbrack {{Eq}.\mspace{14mu} 42} \right\rbrack\end{matrix}$

As before, there is duality between a time and frequency-domainformulation. Let α_(n)=e^(i(μ) ^(2n) ^((φ)−μ) ^(2n−1) ^((φ))). Equation42 may be thought of as either an inner-product or a correlation,φ_(MLE)=argmax_(φ) Re(α*β)   [Eq. 43]Where in the time domain,β_(n) =y* _(2n−1) y _(2n)   [Eq. 2]and in the frequency domainβ_(n) =Y* _(2n−1) Y _(2n).   [Eq. 3]

Consider now a known target signal with multiple pairs of elements.Equation 43 with the associated definitions of α and β is easilyidentified as a generalization of Equations 26 and 28. Similarly,Equations 35 and 36 may also be generalized to Equation 43, where in thetime domain

$\begin{matrix}{\beta_{n} = \frac{\left( {x_{n}^{*}y_{{2n} - 1}} \right)*\left( {x_{n}^{*}y_{2n}} \right)}{{x_{n}}^{2}}} & \left\lbrack {{Eq}.\mspace{14mu} 4} \right\rbrack\end{matrix}$and in the frequency domain

$\begin{matrix}{\beta_{n} = {\frac{\left( {X_{n}^{*}Y_{{2n} - 1}} \right)*\left( {X_{n}^{*}Y_{2n}} \right)}{{X_{n}}^{2}}.}} & \left\lbrack {{Eq}.\mspace{14mu} 5} \right\rbrack\end{matrix}$The subscript n on x and X denotes the known target signal when then^(th) pair is sampled. The appearance of |x_(n)|² and |X_(n)|² in thedenominator is for the reasons discussed previously.Estimation of Unknown Target Signal Bandwidth

Consider now the estimation of the bandwidth of an unknown targetsignal. The estimation process described herein may be used to determinethe boundaries of contiguous spectra, and as such, may also be regardedas a signal detection method. Previously it was demonstrated that, foran unknown target signal,φ_(MLE)=argmax_(φ) Re<α,β>  [Eq. 44]where <α,β> denotes the inner-product of α and β. (<α,β>=α*β.) With afrequency-domain formulation β_(n)=<Y_(2n−1), Y_(2n)>. In general, atarget signal may occupy a bandwidth less than that corresponding to thesampling rate, and some points of the Fourier transforms Y_(2n−1) andY_(2n) may depend only on noise. Intuitively, it may seem that theinner-product for β_(n) should only be computed only over points wherethe target signal may be non-zero. Essentially this turns out to becorrect, but it is also necessary to develop a method for estimation ofthe bandwidth.Recall,∥Y−A(φ)X(φ)∥² =∥Y∥ ² −Y*A(φ)X(φ)   [Eq. 21]One seeks to approximate Y by a judicious choice of A(φ)X(φ) and therebydetermine φ_(MLE). X(φ) is a estimate of the Fourier transform of theunknown target signal. The LHS of Equation 21 is non-negative. Thus,Y*A(φ)X(φ)≦∥Y∥². φ_(MLE) may be determined by maximizing Y*A(φ)X(φ).Recall,

$\begin{matrix}\left. {{{Y*{A(\phi)}{X(\phi)}} = {{c{{{A(\phi)}*Y}}^{2}} = {{{{c\left( {{< Y_{1}},{Y_{1} > +}}\quad \right.}2{Re}\left\{ {{{\mathbb{e}}^{{\mathbb{i}}_{({{\mu_{1}{(\phi)}} - {\mu_{2n}{(\phi)}}})}} < Y_{1}},{Y_{2} >}} \right\}} +} < Y_{2}}}},{Y_{2} >}} \right) & \left\lbrack {{Eq}.\mspace{14mu} 45} \right\rbrack\end{matrix}$This expression is a function of φ. Assuming that e^(i(μ) ¹ ^((φ)−μ) ²^((φ))) ranges over the unit circle,

$\begin{matrix}{{Y*{A\left( \phi_{MLE} \right)}{X\left( \phi_{MLE} \right)}} = {c\left( {\left\langle {Y_{1},Y_{1}} \right\rangle + {2{\left\langle {Y_{1},Y_{2}} \right\rangle }} + \left\langle {Y_{2},Y_{2}} \right\rangle} \right)}} & \left\lbrack {{Eq}.\mspace{14mu} 46} \right\rbrack\end{matrix}$

Consider now the introduction of a bandwidth constraint. Let M denote aset of tuples of X(φ) where X(φ) may be non-zero. It is not necessarilythe case that X(φ) is non-zero for these tuples, but it is assumed thatX(φ) is certainly zero for tuples that are not in the set M. Moreprecisely, in m∉M

X_(m)(φ)=0. Thus M specifies a bandwidth constraint.

Let {tilde over (X)}(φ) denote the vector X(φ) with the zero tuplesdeleted. Thus, {tilde over (X)}(φ) is a |M|-tuple vector. The expressionA(φ)X(φ) is a linear combination of the columns of A(φ), and each columnis weighted by a tuple of X(φ). Similarly, let Ã(φ) denote the matrixA(φ) with the columns that are weighted by zero tuples of X(φ) deleted.In Equations 21, 45 and 46, X(φ) and A(φ) may then be replaced with{tilde over (X)}(φ) and Ã(φ) without consequence, with one exception.The inner-products appearing in Equations 45 and 46 must be computedonly over tuples where X(φ) may be non-zero. Using Equation 45 Equation21 may be rewritten as,

$\begin{matrix}{{{{{{Y - {{\overset{\sim}{A}\left( \phi_{MLE} \right)}{\overset{\sim}{X}\left( \phi_{MLE} \right)}}}}^{2} = {< Y_{1}}},{Y_{1} > {+ {< Y_{2}}}},{Y_{2} > -}}\quad}{c\left( {{< Y_{1}},{Y_{1} >_{M}{{{+ 2}{{{< Y_{1}},{Y_{2} >_{M}}}}} +} < Y_{2}},{Y_{2} >_{M}}} \right)}} & \left\lbrack {{Eq}.\mspace{14mu} 47} \right\rbrack\end{matrix}$The inner-product subscript M denotes a computation over only thenon-zero tuples specified by M, and no subscript indicates the ordinarycomputation over all tuples. Let,E(M)=Y−Ã(φ_(MLE)){tilde over (X)}(φ_(MLE))   [Eq. 48]E(M) is the error between Y and the optimal estimate Ã(φ_(MLE)){tildeover (X)}(φ_(MLE)). M denotes the dependence of the error on thebandwidth constraint. From Equation 47

$\begin{matrix}{{{{{{E(M)}}^{2} = {< Y_{1}}},{Y_{1} > {+ {< Y_{2}}}},{Y_{2} > -}}\quad}{c\left( {{< Y_{1}},{Y_{1} >_{M}{{{+ 2}{{{< Y_{1}},{Y_{2} >_{M}}}}} +} < Y_{2}},{Y_{2} >_{M}}} \right)}} & \left\lbrack {{Eq}.\mspace{14mu} 8} \right\rbrack\end{matrix}$

The bandwidth may be estimated by using multiple hypotheses andgeneralized log likelihood ratio tests (GLLRT). See A. D. Whalen,“Detection of Signals in Noise”, Academic Press, Second Edition, 1995.Suppose that one is deciding between bandwidth hypothesis M₁ and M₂. Theresulting GLLRT is given by,

$\begin{matrix}{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{2} \right)}}^{2}}\underset{M_{2}}{\overset{M_{1}}{\lesseqgtr}}{\lambda\left( {M_{1},M_{2}} \right)}} & \left\lbrack {{Eq}.\mspace{14mu} 7} \right\rbrack\end{matrix}$Where λ(M₁, M₂) is an appropriately chosen constant threshold. Forexample, an appropriate rule in the context of DF is to determine λ(M₁,M₂) by,λ(M ₂ , M ₂)=argmin V ar{φ _(MLE)}  [Eq. 49]Equation 49 is easily solved by simulation. Observe that Equation 49 isa Bayesian criterion, as the cost of error, as measured by the varianceof φ_(MLE), may differ between M₁ and M₂. The extension to more than abinary hypothesis may be accomplished by performing pair wise tests. Inother words,

$\begin{matrix}\left. \begin{matrix}{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{2} \right)}}^{2}} < {\lambda\left( {M_{1},M_{2}} \right)}} \\{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{3} \right)}}^{2}} < {\lambda\left( {M_{1},M_{3}} \right)}}\end{matrix}\Rightarrow M_{1} \right. & \left\lbrack {{Eq}.\mspace{14mu} 9} \right\rbrack\end{matrix}$

The benefits specifically stated herein do not necessarily apply toevery conceivable embodiment of the present invention. Further, suchstated benefits of the present invention are only examples and shouldnot be construed as the only benefits of the present invention.

While the above description contains many specificities, thesespecificities are not to be construed as limitations on the scope of thepresent invention, but rather as examples of the preferred embodimentsdescribed herein. Other variations are possible and the scope of thepresent invention should be determined not by the embodiments describedherein but rather by the claims and their legal equivalents.

Regarding the method claims, except for those steps that can only occurin the sequence in which they are recited, and except for those stepsfor which the occurrence of a given sequence is specifically recited ormust be inferred, the steps of the method claims do not have to occur inthe sequence in which they are recited.

1. A method of estimating the angle of arrival of a target signalreceived by an array of antenna elements, comprising the steps of: (a)with a pair of receivers, simultaneously obtaining samples of a receivedtarget signal from multiple elements of an array of antenna elements;and (b) with a computer, processing the simultaneously obtained samplesof the target signal to determine a maximum likelihood estimation (MLE)of the angle of arrival φ of the target signal by using the followingequation:φ_(MLE)=argmax_(φ) Re(α*β) wherein α is a complex vector that representsa phase difference associated with the angle of arrival that should beobserved upon receipt of the signal by two particular antenna elementsfrom which the samples are obtained; and β represents the phasedifference that is observed upon receipt of the signal by the twoparticular antenna elements from which the samples are obtained; whereinwhen the target signal is unknown, β_(n)=y*_(2n−1)y_(2n) in the timedomain and β_(n)=Y*_(2n−1)Y_(2n) in the frequency domain, whereiny_(2n−1) and y_(2n) are complex N-tuple vectors representing the samplesobtained from the nth simultaneously sampled pair of antenna elementsand Y is a Fourier transform of y.
 2. A method according to claim 1,wherein for a phased array of said antenna elements, α_(n)=e^(i(μ) ^(2n)^((Φ)−μ) ^(2n−1) ^((φ))), wherein μ is a function of the angle ofarrival that depends upon the geometry of the array of antenna elements.3. A method according to claim 1, further comprising the steps of: (c)estimating the bandwidth of the received target signal by using binaryhypotheses and a generalized log likelihood ratio test (GLLRT):${{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{2} \right)}}^{2}}\overset{M_{1}}{\underset{M_{2}}{\lessgtr}}{\lambda\left( {M_{1},M_{2}} \right)}$wherein λ(M₁, M₂) is an appropriately chosen constant threshold; whereinM specifies a bandwidth constraint expressed by a set of tuples of X(φ)where X(φ) may be non-zero, X(φ) is an estimate of the Fourier transformof the unknown target signal, Y is a Fourier transform of the sampleobtained from the sampled antenna element, andE(M)² =  < Y₁, Y₁ > +  < Y₂, Y₂ > −c( < Y₁, Y₁>_(M)+2 < Y₁, Y₂>_(M)+ < Y₂, Y₂>_(M));  and(d) deriving the value of β_(n)=<Y_(2n−1), Y_(2n)>_(M) in the frequencydomain or the value of β_(n)=<y_(2n−1), y_(2n)>_(M) in the time domainby using the respective value of M that is associated with the estimatedbandwidth pursuant to said GLLRT.
 4. A method according to claim 1,further comprising the steps of: (c) estimating the bandwidth of thereceived target signal by using multiple hypotheses and pair wisegeneralized log likelihood ratio tests in accordance with:$\left. \begin{matrix}{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{2} \right)}}^{2}} < {\lambda\left( {M_{1},M_{2}} \right)}} \\{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{3} \right)}}^{2}} < {\lambda\left( {M_{1},M_{3}} \right)}}\end{matrix}\Rightarrow M_{1} \right.$ wherein M specifies a bandwidthconstraint expressed by a set of tuples of X(φ) where X(φ) may benon-zero, X(φ) is an estimate of the Fourier transform of the unknowntarget signal, Y is a Fourier transform of the sample obtained from thesampled antenna element, andE(M)² =  < Y₁, Y₁ > + < Y₂, Y₂ > −             c( < Y₁, Y₁>_(M)+2 < Y₁, Y₂>_(M)+ < Y₂, Y₂>_(M));(d) deriving the value of β_(n)=<Y_(2n−1), Y_(2n)>_(M) in the frequencydomain or the value of β_(n)=<y_(2n−1), y_(2n)>_(M) in the time domainby using the respective value of M that is associated with the estimatedbandwidth pursuant to said generalized log likelihood ratio tests.
 5. Amethod of estimating the angle of arrival of a target signal received byan array of antenna elements, comprising the steps of: (a) with a pairof receivers, simultaneously obtaining samples of a received targetsignal from multiple elements of an array of antenna elements; and (b)with a computer, processing the simultaneously obtained samples of thetarget signal to determine a maximum likelihood estimation (MLE) of theangle of arrival φ of the target signal by using the following equation:φ_(MLE)=argmax_(φ) Re(α*β) wherein α is a complex vector that representsa phase difference associated with the angle of arrival that should beobserved upon receipt of the signal by two particular antenna elementsfrom which the samples are obtained; and β represents the phasedifference that is observed upon receipt of the signal by the twoparticular antenna elements from which the samples are obtained; whereinwhen the target signal is known,$\beta_{n} = \frac{\left( {x_{n}^{*}y_{{2n} - 1}} \right)*\left( {x_{n}^{*}y_{2n}} \right)}{{x_{n}}^{2}}$in the time domain and$\beta_{n} = \frac{\left( {X_{n}^{*}Y_{{2n} - 1}} \right)*\left( {X_{n}^{*}Y_{2n}} \right)}{{X_{n}}^{2}}$in the frequency domain, wherein X and Y are Fourier transforms of x andy respectively, wherein x_(n) and X_(n) are complex N-tuple vectorsrepresenting the known target signal in the time domain and in thefrequency domain respectively; and wherein y_(2n−1) and y_(2n) arecomplex N-tuple vectors representing the samples obtained from the nthsimultaneously sampled pair of antenna elements and Y is a Fouriertransform of y.
 6. A method according to claim 5, wherein for a phasedarray of said antenna elements, α_(n)=e^(i(μ) ^(2n) ^((φ)−μ) ^(2n−1)^((φ)),) wherein μ is a function of the angle of arrival that dependsupon the geometry of the array of antenna elements.
 7. A method ofestimating the bandwidth of a target signal received by an array ofantenna elements, comprising the steps of: (a) with a pair of receivers,simultaneously obtaining samples of a received target signal frommultiple elements of an array of antenna elements; and (b) with acomputer, processing the simultaneously obtained samples of the targetsignal to estimate the bandwidth of the received target signal by usingbinary hypotheses and a generalized log likelihood ratio test (GLLRT).8. A method according to claim 7, wherein the GLLRT is:${{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{2} \right)}}^{2}}\overset{M_{1}}{\underset{M_{2}}{\lessgtr}}{\lambda\left( {M_{1},M_{2}} \right)}$wherein λ(M₁, M₂) is an appropriately chosen constant threshold; whereinM specifies a bandwidth constraint expressed by a set of tuples of X(φ)where X(φ) may be non-zero, X(φ) is an estimate of the Fourier transformof the unknown target signal, Y is a Fourier transform of the sampleobtained from the sampled antenna element, andE(M)² =  < Y₁, Y₁ > +      < Y₂, Y₂ > −c( < Y₁, Y₁>_(M)+2 < Y₁, Y₂>_(M)+ < Y₂, Y₂>_(M)).9. A method according to claim 8, wherein λ(M₁, M₂)=argmin Var{φ_(MLE)}.
 10. A method of estimating the bandwidth of a target signalreceived by an array of antenna elements, comprising the steps of: (a)with a pair of receivers, simultaneously obtaining samples of a receivedtarget signal from multiple elements of an array of antenna elements;and (b) with a computer, processing the simultaneously obtained samplesof the target signal to estimate the bandwidth of the received targetsignal by using multiple hypotheses and pair wise generalized loglikelihood ratio tests in accordance with: $\left. \begin{matrix}{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{2} \right)}}^{2}} < {\lambda\left( {M_{1},M_{2}} \right)}} \\{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{3} \right)}}^{2}} < {\lambda\left( {M_{1},M_{3}} \right)}}\end{matrix}\Rightarrow M_{1} \right.$ wherein M specifies a bandwidthconstraint expressed by a set of tuples of X(φ) where X(φ) may benon-zero, X(φ) is an estimate of the Fourier transform of the unknowntarget signal, Y is a Fourier transform of the sample obtained from thesampled antenna element, andE(M)² =  < Y₁, Y₁ > +      < Y₂, Y₂ > −c( < Y₁, Y₁>_(M)+2 < Y₁, Y₂>_(M)+ < Y₂, Y₂>_(M)).11. A system for estimating the angle of arrival of a target signalreceived by an array of antenna elements, comprising: a pair ofreceivers adapted for simultaneously obtaining samples of a receivedtarget signal from multiple elements of an array of antenna elements;and a computer adapted for processing the simultaneously obtainedsamples of the target signal to determine a maximum likelihoodestimation (MLE) of the angle of arrival φ of the target signal by usingthe following equation:φ_(MLE)=argmax_(φ) Re(α*β) wherein α is a complex vector that representsa phase difference associated with the angle of arrival that should beobserved upon receipt of the signal by two particular antenna elementsfrom which the samples are obtained; and β represents the phasedifference that is observed upon receipt of the signal by the twoparticular antenna elements from which the samples are obtained; whereinwhen the target signal is unknown, β_(n)=y*_(2n−1)y_(2n) in the timedomain and β_(n)=Y*_(2n−1)Y_(2n) in the frequency domain, whereiny_(2n−1) and y_(2n) are complex N-tuple vectors representing the samplesobtained from the nth simultaneously sampled pair of antenna elementsand Y is a Fourier transform of y.
 12. A system according to claim 11,further comprising: means for estimating the bandwidth of the receivedtarget signal by using binary hypotheses and a generalized loglikelihood ratio test (GLLRT); and means for deriving the value ofβ_(n)=<Y_(2n−1), Y_(2n)>_(M) in the frequency domain or the value ofβ_(n)=<y_(2n−1), y_(2n)>_(M) in the time domain by using the respectivevalue of M that is associated with the estimated bandwidth pursuant tosaid GLLRT.
 13. A system according to claim 11, further comprising thesteps of: means for estimating the bandwidth of the received targetsignal by using multiple hypotheses and pair wise generalized loglikelihood ratio tests in accordance with: $\left. \begin{matrix}{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{2} \right)}}^{2}} < {\lambda\left( {M_{1},M_{2}} \right)}} \\{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{3} \right)}}^{2}} < {\lambda\left( {M_{1},M_{3}} \right)}}\end{matrix}\Rightarrow M_{1} \right.$ wherein M specifies a bandwidthconstraint expressed by a set of tuples of X(φ) where X(φ) may benon-zero, X(φ) is an estimate of the Fourier transform of the unknowntarget signal, Y is a Fourier transform of the sample obtained from thesampled antenna element, andE(M)² =  < Y₁, Y₁ > +      < Y₂, Y₂ > −c( < Y₁, Y₁>_(M)+2 < Y₁, Y₂>_(M)+ < Y₂, Y₂>_(M));means for deriving the value of β_(n)=<Y_(2n−1), Y_(2n)>_(M) in thefrequency domain or the value of β_(n)=<y_(2n−1), y_(2n)>_(M) in thetime domain by using the respective value of M that is associated withthe estimated bandwidth pursuant to said generalized log likelihoodratio tests.
 14. A system for estimating the angle of arrival of atarget signal received by an array of antenna elements, comprising thesteps of: a pair of receivers adapted for simultaneously obtainingsamples of a received target signal from multiple elements of an arrayof antenna elements; and a computer adapted for processing thesimultaneously obtained samples of the target signal to determine amaximum likelihood estimation (MLE) of the angle of arrival φ of thetarget signal by using the following equation:φ_(MLE)=argmax_(φ) Re(α*β) wherein α is a complex vector that representsa phase difference associated with the angle of arrival that should beobserved upon receipt of the signal by two particular antenna elementsfrom which the samples are obtained; and β represents the phasedifference that is observed upon receipt of the signal by the twoparticular antenna elements from which the samples are obtained; whereinwhen the target signal is known,$\beta_{n} = \frac{\left( {x_{n}^{*}y_{{2n} - 1}} \right)*\left( {x_{n}^{*}y_{2n}} \right)}{{x_{n}}^{2}}$in the time domain and$\beta_{n} = \frac{\left( {X_{n}^{*}Y_{{2n} - 1}} \right)*\left( {X_{n}^{*}Y_{2n}} \right)}{{X_{n}}^{2}}$in the frequency domain, wherein X and Y are Fourier transforms of x andy respectively, wherein x_(n) and X_(n) are complex N-tuple vectorsrepresenting the known target signal in the time domain and in thefrequency domain respectively; and wherein y_(2n−1) and y_(2n) arecomplex N-tuple vectors representing the samples obtained from the nthsimultaneously sampled pair of antenna elements and Y is a Fouriertransform of y.
 15. A system for estimating the bandwidth of a targetsignal received by an array of antenna elements, comprising the stepsof: a pair of receivers adapted for simultaneously obtaining samples ofa received target signal from multiple elements of an array of antennaelements; and a computer adapted for processing the simultaneouslyobtained samples of the target signal to estimate the bandwidth of thereceived target signal by using binary hypotheses and a generalized loglikelihood ratio test (GLLRT).
 16. A system according to claim 15,wherein the GLLRT is:${{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{2} \right)}}^{2}}\underset{M_{2}}{\overset{M_{1}}{\lessgtr}}{\lambda\left( {M_{1},M_{2}} \right)}$wherein λ(M₁, M₂) is an appropriately chosen constant threshold; whereinM specifies a bandwidth constraint expressed by a set of tuples of X(φ)where X(φ) may be non-zero, X(φ) is an estimate of the Fourier transformof the unknown target signal, Y is a Fourier transform of the sampleobtained from the sampled antenna element, andE(M)² = ⟨Y₁, Y₁⟩ + ⟨Y₂, Y₂⟩ − c(⟨Y₁, Y₁⟩_(M) + 2⟨Y₁, Y₂⟩_(M) + ⟨Y₂, Y₂⟩_(M)).17. A system according to claim 16, wherein λ(M₁, M₂)=argmin Var{φ_(MLE)}.
 18. A system for estimating the bandwidth of a targetsignal received by an array of antenna elements, comprising the stepsof: a pair of receivers adapted for simultaneously obtaining samples ofa received target signal from multiple elements of an array of antennaelements; and a computer adapted for processing the simultaneouslyobtained samples of the target signal to estimate the bandwidth of thereceived target signal by using multiple hypotheses and pair wisegeneralized log likelihood ratio tests in accordance with:$\left. \begin{matrix}{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{2} \right)}}^{2}} < {\lambda\left( {M_{1},M_{2}} \right)}} \\{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{3} \right)}}^{2}} < {\lambda\left( {M_{1},M_{3}} \right)}}\end{matrix}\Rightarrow M_{1} \right.$ wherein M specifies a bandwidthconstraint expressed by a set of tuples of X(φ) where X(φ) may benon-zero, X(φ) is an estimate of the Fourier transform of the unknowntarget signal, Y is a Fourier transform of the sample obtained from thesampled antenna element, andE(M)² = ⟨Y₁, Y₁⟩ + ⟨Y₂, Y₂⟩ − c(⟨Y₁, Y₁⟩_(M) + 2⟨Y₁, Y₂⟩_(M) + ⟨Y₂, Y₂⟩_(M)).19. A computer readable storage medium for use with a computer in asystem for determining the angle of arrival of a target signal receivedby an array of antenna elements, wherein the system comprises: a pair ofreceivers adapted for simultaneously obtaining samples of a receivedtarget signal from multiple elements of an array of antenna elements;and a computer, wherein the computer readable storage medium containscomputer executable program instructions for causing the computer toprocess the simultaneously obtained samples of the target signal todetermine a maximum likelihood estimation (MLE) of the angle of arrivalφ of the target signal by using the following equation:φ_(MLE)=argmax_(φ) Re(α*β) wherein α is a complex vector that representsa phase difference associated with the angle of arrival that should beobserved upon receipt of the signal by two particular antenna elementsfrom which the samples are obtained; and β represents the phasedifference that is observed upon receipt of the signal by the twoparticular antenna elements from which the samples are obtained; whereinwhen the target signal is unknown, β_(n)=y*_(2n−1)y_(2n) in the timedomain and β_(n)=Y*_(2n−1)Y_(2n) in the frequency domain, whereiny_(2n−1) and y_(2n) are complex N-tuple vectors representing the samplesobtained from the nth simultaneously sampled pair of antenna elementsand Y is a Fourier transform of y.
 20. A computer readable storagemedium according to claim 19, wherein for a phased array of said antennaelements, α_(n)=e^(i(μ) ^(2n) ^((φ)−μ) ^(2n−1) ^((φ))), wherein μ is afunction of the angle of arrival that depends upon the geometry of thearray of antenna elements.
 21. A computer readable storage mediumaccording to claim 19, further comprising the steps of: (c) estimatingthe bandwidth of the received target signal by using binary hypothesesand a generalized log likelihood ratio test (GLLRT):${{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{2} \right)}}^{2}}\underset{M_{2}}{\overset{M_{1}}{\lessgtr}}{\lambda\left( {M_{1},M_{2}} \right)}$wherein λ(M₁, M₂) is an appropriately chosen constant threshold; whereinM specifies a bandwidth constraint expressed by a set of tuples of X(φ)where X(φ) may be non-zero, X(φ) is an estimate of the Fourier transformof the unknown target signal, Y is a Fourier transform of the sampleobtained from the sampled antenna element, andE(M)² = ⟨Y₁, Y₁⟩ + ⟨Y₂, Y₂⟩ − c(⟨Y₁, Y₁⟩_(M) + 2⟨Y₁, Y₂⟩_(M) + ⟨Y₂, Y₂⟩_(M)); and(d) deriving the value of β_(n)=<Y_(2n−1), Y_(2n)>_(M) in the frequencydomain or the value of β_(n)=<y_(2n−1), y_(2n)>_(M) in the time domainby using the respective value of M that is associated with the estimatedbandwidth pursuant to said GLLRT.
 22. A computer readable storage mediumaccording to claim 19, further comprising the steps of: (c) estimatingthe bandwidth of the received target signal by using multiple hypothesesand pair wise generalized log likelihood ratio tests in accordance with:$\left. \begin{matrix}{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{2} \right)}}^{2}} < {\lambda\left( {M_{1},M_{2}} \right)}} \\{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{3} \right)}}^{2}} < {\lambda\left( {M_{1},M_{3}} \right)}}\end{matrix}\Rightarrow M_{1} \right.$ wherein M specifies a bandwidthconstraint expressed by a set of tuples of X(φ) where X(φ) may benon-zero, X(φ) is an estimate of the Fourier transform of the unknowntarget signal, Y is a Fourier transform of the sample obtained from thesampled antenna element, andE(M)² = ⟨Y₁, Y₁⟩ + ⟨Y₂, Y₂⟩ − c(⟨Y₁, Y₁⟩_(M) + 2⟨Y₁, Y₂⟩_(M) + ⟨Y₂, Y₂⟩_(M));(d) deriving the value of β_(n)=<Y_(2n−1), Y_(2n)>_(M) in the frequencydomain or the value of β_(n)=<y_(2n−1), y_(2n)>_(M) in the time domainby using the respective value of M that is associated with the estimatedbandwidth pursuant to said generalized log likelihood ratio tests.
 23. Acomputer readable storage medium for use with a computer in a system fordetermining the angle of arrival of a target signal received by an arrayof antenna elements, wherein the system comprises: a pair of receiversadapted for simultaneously obtaining samples of a received target signalfrom multiple elements of an array of antenna elements; and a computer,wherein the computer readable storage medium contains computerexecutable program instructions for causing the computer to process thesimultaneously obtained samples of the target signal to determine amaximum likelihood estimation (MLE) of the angle of arrival φ of thetarget signal by using the following equation:φ_(MLE)=argmax_(φ) Re(α*β) wherein α is a complex vector that representsa phase difference associated with the angle of arrival that should beobserved upon receipt of the signal by two particular antenna elementsfrom which the samples are obtained; and β represents the phasedifference that is observed upon receipt of the signal by the twoparticular antenna elements from which the samples are obtained; whereinwhen the target signal is known,$\beta_{n} = \frac{\left( {x_{n}^{*}y_{{2n} - 1}} \right)*\left( {x_{n}^{*}y_{2n}} \right)}{{x_{n}}^{2}}$in the time domain and$\beta_{n} = \frac{\left( {X_{n}^{*}Y_{{2n} - 1}} \right)*\left( {X_{n}^{*}Y_{2n}} \right)}{{X_{n}}^{2}}$in the frequency domain, wherein X and Y are Fourier transforms of x andy respectively, wherein x_(n) and X_(n) are complex N-tuple vectorsrepresenting the known target signal in the time domain and in thefrequency domain respectively; and wherein Y_(2n−1) and y_(2n) arecomplex N-tuple vectors representing the samples obtained from the nthsimultaneously sampled pair of antenna elements and Y is a Fouriertransform of y.
 24. A computer readable storage medium according toclaim 23, wherein for a phased array of said antenna elements,α_(n)=e^(i(μ) ^(2n) ^((φ)−μ) ^(2n−1) ^((φ))), wherein μ is a function ofthe angle of arrival that depends upon the geometry of the array ofantenna elements.
 25. A computer readable storage medium for use with acomputer in a system for estimating the bandwidth of a target signalreceived by an array of antenna elements, wherein the system comprises:a pair of receivers adapted for simultaneously obtaining samples of areceived target signal from multiple elements of an array of antennaelements; and a computer, wherein the computer readable storage mediumcontains computer executable program instructions for causing thecomputer to process the simultaneously obtained samples of the targetsignal to estimate the bandwidth of the received target signal by usingbinary hypotheses and a generalized log likelihood ratio test (GLLRT).26. A computer readable storage medium according to claim 25, whereinthe GLLRT is:${{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{2} \right)}}^{2}}\underset{M_{2}}{\overset{M_{1}}{\lessgtr}}{\lambda\left( {M_{1},M_{2}} \right)}$wherein λ(M₁ , M₂) is an appropriately chosen constant threshold;wherein M specifies a bandwidth constraint expressed by a set of tuplesof X(φ) where X(φ) may be non-zero, X(φ) is an estimate of the Fouriertransform of the unknown target signal, Y is a Fourier transform of thesample obtained from the sampled antenna element, andE(M)² = ⟨Y₁, Y₁⟩ + ⟨Y₂, Y₂⟩ − c(⟨Y₁, Y₁⟩_(M) + 2⟨Y₁, Y₂⟩_(M) + ⟨Y₂, Y₂⟩_(M)).27. A computer readable storage medium according to claim 26, whereinλ(M₁, M₂)=argmin V ar{φ_(MLE)}.
 28. A computer readable storage mediumfor use with a computer in a system for estimating the bandwidth of atarget signal received by an array of antenna elements, wherein thesystem comprises: a pair of receivers adapted for simultaneouslyobtaining samples of a received target signal from multiple elements ofan array of antenna elements; and a computer, wherein the computerreadable storage medium contains computer executable programinstructions for causing the computer to process the simultaneouslyobtained samples of the target signal to estimate the bandwidth of thereceived target signal by using multiple hypotheses and pair wisegeneralized log likelihood ratio tests in accordance with:$\left. \begin{matrix}{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{2} \right)}}^{2}} < {\lambda\left( {M_{1},M_{2}} \right)}} \\{{{{E\left( M_{1} \right)}}^{2} - {{E\left( M_{3} \right)}}^{2}} < {\lambda\left( {M_{1},M_{3}} \right)}}\end{matrix}\Rightarrow M_{1} \right.$ wherein M specifies a bandwidthconstraint expressed by a set of tuples of X(φ) where X(φ) may benon-zero, X(φ) is an estimate of the Fourier transform of the unknowntarget signal, Y is a Fourier transform of the sample obtained from thesampled antenna element, andE(M)² = ⟨Y₁, Y₁⟩ + ⟨Y₂, Y₂⟩ − c(⟨Y₁, Y₁⟩_(M) + 2⟨Y₁, Y₂⟩_(M) + ⟨Y₂, Y₂⟩_(M)).